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2007 AMC 10B problems

All 25 problems from the 2007 AMC 10B, with answer choices, worked solutions and hints. Problems are roughly ordered by difficulty: 1–10 are the most approachable, 19–25 the hardest.

Problems

  1. 1Problem 1Isabella’s house has 3 bedrooms. Each bedroom is 12 feet long, 10 feet wide, and 8 feet high. Isabella must paint the walls of all the bedrooms…Geometry
  2. 2Problem 2Define the operation ★ by a★ b = (a+b)b. What is (3★ 5)-(5★ 3)?Algebra
  3. 3Problem 3A college student drove his compact car 120 miles home for the weekend and averaged 30 miles per gallon. On the return trip the student drove his…Algebra
  4. 4Problem 4The point O is the center of the circle circumscribed about △ ABC, with ∠ BOC = 120° and ∠ AOB = 140°, as shown. What is the degree measure of ∠ ABC?Geometry
  5. 5Problem 5In a certain land, all Arogs are Brafs, all Crups are Brafs, all Dramps are Arogs, and all Crups are Dramps. Which of the following statements is…
  6. 6Problem 6The 2007 AMC 10 will be scored by awarding 6 points for each correct response, 0 points for each incorrect response, and 1.5 points for each problem…Algebra
  7. 7Problem 7All sides of the convex pentagon ABCDE are of equal length, and ∠ A = ∠ B = 90°. What is the degree measure of ∠ E?Geometry
  8. 8Problem 8On the trip home from the meeting where this AMC 10 was constructed, the Contest Chair noted that his airport parking receipt had digits of the form…Number Theory
  9. 9Problem 9A cryptographic code is designed as follows. The first time a letter appears in a given message it is replaced by the letter that is 1 place to its…Algebra
  10. 10Problem 10Two points B and C are in a plane. Let S be the set of all points A in the plane for which △ ABC has area 1. Which of the following describes S?Geometry
  11. 11Problem 11A circle passes through the three vertices of an isosceles triangle that has two sides of length 3 and a base of length 2. What is the area of this…Geometry
  12. 12Problem 12Tom’s age is T years, which is also the sum of the ages of his three children. His age N years ago was twice the sum of their ages then. What is T/N?Algebra
  13. 13Problem 13Two circles of radius 2 are centered at (2,0) and at (0,2). What is the area of the intersection of the interiors of the two circles?Geometry
  14. 14Problem 14Some boys and girls are having a car wash to raise money for a class trip to China. Initially 40% of the group are girls. Shortly thereafter two…Algebra
  15. 15Problem 15The angles of quadrilateral ABCD satisfy ∠ A = 2∠ B = 3∠ C = 4∠ D. What is the degree measure of ∠ A, rounded to the nearest whole number?Algebra
  16. 16Problem 16A teacher gave a test to a class in which 10% of the students are juniors and 90% are seniors. The average score on the test was 84. The juniors all…Algebra
  17. 17Problem 17Point P is inside equilateral △ ABC. Points Q, R, and S are the feet of the perpendiculars from P to AB, BC, and CA, respectively. Given that PQ = 1…Geometry
  18. 18Problem 18A circle of radius 1 is surrounded by 4 circles of radius r as shown. What is r?Geometry
  19. 19Problem 19The wheel shown is spun twice, and the randomly determined numbers opposite the pointer are recorded. The first number is divided by 4, and the…Number Theory
  20. 20Problem 20A set of 25 square blocks is arranged into a 5× 5 square. How many different combinations of 3 blocks can be selected from that set so that no two…Counting & Probability
  21. 21Problem 21Right △ ABC has AB=3, BC=4, and AC=5. Square XYZW is inscribed in △ ABC with X and Y on AC, W on AB, and Z on BC. What is the side length of the…Geometry
  22. 22Problem 22A player chooses one of the numbers 1 through 4. After the choice has been made, two regular four-sided (tetrahedral) dice are rolled, with the sides…Counting & Probability
  23. 23Problem 23A pyramid with a square base is cut by a plane that is parallel to its base and is 2 units from the base. The surface area of the smaller pyramid…Geometry
  24. 24Problem 24Let n denote the smallest positive integer that is divisible by both 4 and 9, and whose base-10 representation consists of only 4’s and 9’s, with at…Number Theory
  25. 25Problem 25How many pairs of positive integers (a,b) are there such that a and b have no common factors greater than 1 and a/b+14b/9a is an integer?Number Theory

Practise the same ideas across every year on the topic pages, or browse the full AMC 10 archive.